32 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354
This problem depends upon one of Fourier's solutions which is easily verified*. We have
_ _ _ __!__ e-xPfivt i T A\
" ' ........................
F*-* ' I _y> 7 /I T \ ----— e 3 a^.........................(11)
V"" Jo
For the reaction on the plane we require only the value of dv/das when x = 0. And
= - 77^............................(12)
Stokes continues f "now suppose the plane to be moved in any manner, so that its velocity at the end of the time t is V (t). We may evidently obtain the result in this case by writing V (T) dr for V, and t — T for t in [12], and integrating with respect to T. We thus get
v'dr i r <y ,
l '' ">u '
and since T3 = — /j,dvfdx0, these formulas solve the problem of finding the reaction in the general case.
There is another method by which the present problem may be treated, and a comparison leads to a transformation which we shall find useful further on. Starting from the periodic solution (8), we may generalize it by Fourier's theorem. Thus
n/v)dn......................(14)
'o corresponds to
7(0= [ Vnetntdn, ..........................(15)
where Vn is an arbitrary function of n. Comparing (13) and (14), we see that
rf V (T) dr
. j?t—~\................(16)
It is easy to verify (16). If we substitute on the right for F'(T) from (15), we get
1 I'* dr r°° _ .
and taking first the integration with respect to r,
when (16) follows at once.
* Compare Kelvin, Ed. Trans. 1862 ; Thomson and Tait, Appendix D. t I have made some small changes of notation. same effect as increasing the inertia of the plane." It will be observed that if Vn be given, the force diminishes without limit with n.